Planar structure for inclusions of finite von Neumann algebras
This dissertation consists of three self-contained papers from my graduate work at UC Berkeley. The chapters increase in complexity from the annular Temperley-Lieb category to strongly Markov inclusions of finite von Neumann algebras to infinite index II_1-subfactors.
In Chapter 2, we discuss how two copies of the cyclic category generate the annular Temperley-Lieb category. In the process, we give a presentation of the annular Temperley-Lieb category via generators and relations, and we see the cyclic category evolve from the simplicial and semi-simplicial categories.
Chapter 3 is joint work with Vaughan F. R. Jones. First, we define a canonical planar *-algebra associated to a strongly Markov inclusion of finite von Neumann algebras (the notion of such an inclusion is defined within). Second, we show for an inclusion of finite dimensional C^*-algebras with the Markov trace, the canonical planar algebra is isomorphic to the graph planar algebra of the Bratteli diagram of the inclusion. We use this fact to show that a subfactor planar algebra embeds into the graph planar algebra of its principal graph.
In Chapter 4, we expand upon Burns' work on rotations for infinite index II_1-subfactors. We start with a II_1-factor bimodule, and we construct a tower of centralizer algebras and a sequence of central L^2-vectors. In the finite index setting, the centralizer algebras and central L^2-vectors agree, but in the infinite index setting, these spaces can differ dramatically. We develop planar calculi for both sequences which are compatible. Interestingly, we obtain planar structure without Jones' basic construction or the resulting Jones projections! We also generalize Burns work on extremality and the existence of rotations to the bimodule setting, and we recover his main theorem. Along the way, we prove some results about relative tensor products of extended positive cones, and we give an example of an infinite index subfactor with finite dimensional higher relative commutants.